For what values of x is $f(x)=((5x)/2)^(2/3) - x^(5/3$ concave or convex?

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Answer 1 · 2018-06-16T14:03:04

$f(x)$ is concave for all $x$ with $x>0$

Writing $f(x)$ in the form

$f(x)=(5/2)^(2/3)*x^(2/3)-x^(5/3)$
then

$f'(x)=(5/2)^(2/3)*2/3*x^(-1/3)-5/3*x^(2/3)$
so
$f''(x)=(5/3)^(2/3)*(-2/9)*x^(-4/3)-10/9*x^(-1/3)<0$

For what values of x is f(x)=((5x)/2)^(2/3) - x^(5/3f(x)=((5x)/2)^(2/3) - x^(5/3 concave or convex?